December 2018, 11(6): 1395-1426. doi: 10.3934/krm.2018055

Numerical study of an anisotropic Vlasov equation arising in plasma physics

Université de Toulouse & CNRS, UPS, Institut de Mathématiques de Toulouse UMR 5219, F-31062, Toulouse, France

* Corresponding author: C. Negulescu

Received  October 2016 Revised  July 2017 Published  June 2018

Goal of this paper is to investigate several numerical schemes for the resolution of two anisotropic Vlasov equations. These two toy-models are obtained from a kinetic description of a tokamak plasma confined by strong magnetic fields. The simplicity of our toy-models permits to better understand the features of each scheme, in particular to investigate their asymptotic-preserving properties, in the aim to choose then the most adequate numerical scheme for upcoming, more realistic simulations.

Citation: Baptiste Fedele, Claudia Negulescu. Numerical study of an anisotropic Vlasov equation arising in plasma physics. Kinetic & Related Models, 2018, 11 (6) : 1395-1426. doi: 10.3934/krm.2018055
References:
[1]

M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, Journal of Differential Equations, 249 (2010), 1620-1663. doi: 10.1016/j.jde.2010.07.010.

[2]

F. F Chen, Plasma Physics and Controlled Fusion, 3rd edition, Springer-Verlag, New-York, 2008.

[3]

N. Crouseilles and M. Lemou, An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits, Kinetic Related Models, 4 (2011), 441-477. doi: 10.3934/krm.2011.4.441.

[4]

N. CrouseillesM. Lemou and F. Méhats, Asymptotic-Preserving schemes for oscillatory Vlasov-Poisson equations, Journal of Computational Physics, 248 (2013), 287-308. doi: 10.1016/j.jcp.2013.04.022.

[5]

A. De CeccoC. Negulescu and S. Possanner, Asymptotic transition from kinetic to adiabatic electrons along magnetic field lines, SIAM MMS (Multiscale Model. Simul.), 15 (2017), 309-338. doi: 10.1137/15M1043686.

[6]

P. DegondF. DeluzetA. LozinskiJ. Narski and C. Negulescu, Duality based asymptotic-preserving method for highly anisotropic diffusion equations, Communications in Mathematical Sciences, 10 (2012), 1-31. doi: 10.4310/CMS.2012.v10.n1.a2.

[7]

P. DegondA. LozinskiJ. Narski and C. Negulescu, An Asymptotic-Preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition, Journal of Computational Physics, 231 (2012), 2724-2740. doi: 10.1016/j.jcp.2011.11.040.

[8]

F. Filbet and S. Jin, An Asymptotic Preserving scheme for the ES-BGK model of the Boltzmann equation, J. Sci. Computing, 46 (2011), 204-224. doi: 10.1007/s10915-010-9394-x.

[9]

X. GarbetY. IdomuraL. Villard and T. Watanabe, Gyrokinetic simulations of turbulent transport, Nuclear Fusion, 50 (2010). doi: 10.1088/0029-5515/50/4/043002.

[10]

Ph. GhendrihM. Hauray and A. Nouri, Derivation of a gyrokinetic model, existence and uniqueness of specific stationary solutions, Kinetic and Related Models, 2 (2009), 707-725. doi: 10.3934/krm.2009.2.707.

[11]

R. J. Goldston and P. H. Rutherford, Plasma Physics, Taylor & Francis Group, Boca Raton, 1995.

[12]

V. Grandgirard, Y. Sarazin, X. Garbet, G. Dif-Pradalier, Ph. Ghendrih, N. Crouseilles, G. Latu, E. Sonnendrücker, N. Besse and P. Bertrand, GYSELA, a full-f global gyrokinetic semi-lagrangian code for ITG turbulence simulations, Theory of Fusion Plasmas, 871 (2006), American Institute of Physics Conference Series, 100–111.

[13]

R. D. Hazeltine and J. D. Meiss, Plasma Confinement, Dover Publications, New York, 2003.

[14]

M. H. Holmes, Introduction to Numerical Methods in Differential Equations, Springer-Verlag, New York, 2007. doi: 10.1007/978-0-387-68121-4.

[15]

S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review, Rivista di Matematica della Universita di Parma, 3 (2012), 177-216.

[16]

M. Lemou and L. Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 31 (2008), 334-368. doi: 10.1137/07069479X.

[17]

R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, Philadelphia, 2007. doi: 10.1137/1.9780898717839.

[18]

A. LozinskiJ. Narski and C. Negulescu, Highly anisotropic temperature balance equation and its asymptotic-preserving resolution, M2AN (Mathematical Modelling and Numerical Analysis), 48 (2014), 1701-1724. doi: 10.1051/m2an/2014016.

[19]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002.

[20]

A. Mentrelli and C. Negulescu, Asymptotic-Preserving scheme for highly anisotropic non-linear diffusion equations, Journal of Comp. Phys, 231 (2012), 8229-8245. doi: 10.1016/j.jcp.2012.08.004.

[21]

C. Negulescu, Kinetic modelling of strongly magnetized tokamak plasmas with mass disparate particles, the electron Boltzmann relation, submitted.

[22]

C. Negulescu, Asymptotic-Preserving schemes. Modeling, simulation and mathematical analysis of magnetically confined plasmas, Rivista di Matematica della Universita di Parma, 4 (2013), 265-343.

[23]

L. Trefethen and D. Bau, Numerical Linear Algebra, SIAM, Philadelphia, 1997. doi: 10.1137/1.9780898719574.

show all references

References:
[1]

M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, Journal of Differential Equations, 249 (2010), 1620-1663. doi: 10.1016/j.jde.2010.07.010.

[2]

F. F Chen, Plasma Physics and Controlled Fusion, 3rd edition, Springer-Verlag, New-York, 2008.

[3]

N. Crouseilles and M. Lemou, An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits, Kinetic Related Models, 4 (2011), 441-477. doi: 10.3934/krm.2011.4.441.

[4]

N. CrouseillesM. Lemou and F. Méhats, Asymptotic-Preserving schemes for oscillatory Vlasov-Poisson equations, Journal of Computational Physics, 248 (2013), 287-308. doi: 10.1016/j.jcp.2013.04.022.

[5]

A. De CeccoC. Negulescu and S. Possanner, Asymptotic transition from kinetic to adiabatic electrons along magnetic field lines, SIAM MMS (Multiscale Model. Simul.), 15 (2017), 309-338. doi: 10.1137/15M1043686.

[6]

P. DegondF. DeluzetA. LozinskiJ. Narski and C. Negulescu, Duality based asymptotic-preserving method for highly anisotropic diffusion equations, Communications in Mathematical Sciences, 10 (2012), 1-31. doi: 10.4310/CMS.2012.v10.n1.a2.

[7]

P. DegondA. LozinskiJ. Narski and C. Negulescu, An Asymptotic-Preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition, Journal of Computational Physics, 231 (2012), 2724-2740. doi: 10.1016/j.jcp.2011.11.040.

[8]

F. Filbet and S. Jin, An Asymptotic Preserving scheme for the ES-BGK model of the Boltzmann equation, J. Sci. Computing, 46 (2011), 204-224. doi: 10.1007/s10915-010-9394-x.

[9]

X. GarbetY. IdomuraL. Villard and T. Watanabe, Gyrokinetic simulations of turbulent transport, Nuclear Fusion, 50 (2010). doi: 10.1088/0029-5515/50/4/043002.

[10]

Ph. GhendrihM. Hauray and A. Nouri, Derivation of a gyrokinetic model, existence and uniqueness of specific stationary solutions, Kinetic and Related Models, 2 (2009), 707-725. doi: 10.3934/krm.2009.2.707.

[11]

R. J. Goldston and P. H. Rutherford, Plasma Physics, Taylor & Francis Group, Boca Raton, 1995.

[12]

V. Grandgirard, Y. Sarazin, X. Garbet, G. Dif-Pradalier, Ph. Ghendrih, N. Crouseilles, G. Latu, E. Sonnendrücker, N. Besse and P. Bertrand, GYSELA, a full-f global gyrokinetic semi-lagrangian code for ITG turbulence simulations, Theory of Fusion Plasmas, 871 (2006), American Institute of Physics Conference Series, 100–111.

[13]

R. D. Hazeltine and J. D. Meiss, Plasma Confinement, Dover Publications, New York, 2003.

[14]

M. H. Holmes, Introduction to Numerical Methods in Differential Equations, Springer-Verlag, New York, 2007. doi: 10.1007/978-0-387-68121-4.

[15]

S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review, Rivista di Matematica della Universita di Parma, 3 (2012), 177-216.

[16]

M. Lemou and L. Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 31 (2008), 334-368. doi: 10.1137/07069479X.

[17]

R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, Philadelphia, 2007. doi: 10.1137/1.9780898717839.

[18]

A. LozinskiJ. Narski and C. Negulescu, Highly anisotropic temperature balance equation and its asymptotic-preserving resolution, M2AN (Mathematical Modelling and Numerical Analysis), 48 (2014), 1701-1724. doi: 10.1051/m2an/2014016.

[19]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002.

[20]

A. Mentrelli and C. Negulescu, Asymptotic-Preserving scheme for highly anisotropic non-linear diffusion equations, Journal of Comp. Phys, 231 (2012), 8229-8245. doi: 10.1016/j.jcp.2012.08.004.

[21]

C. Negulescu, Kinetic modelling of strongly magnetized tokamak plasmas with mass disparate particles, the electron Boltzmann relation, submitted.

[22]

C. Negulescu, Asymptotic-Preserving schemes. Modeling, simulation and mathematical analysis of magnetically confined plasmas, Rivista di Matematica della Universita di Parma, 4 (2013), 265-343.

[23]

L. Trefethen and D. Bau, Numerical Linear Algebra, SIAM, Philadelphia, 1997. doi: 10.1137/1.9780898719574.

Figure 1.  Representation of the initial condition $f_{in}$ (A) and the exact solution $f_{ex}^{\epsilon}$ at the final time $T = 1$ (B). Here $\epsilon = 1$.
Figure 2.  Representation of the exact limit solution $f^0_{ex}(t,x)$ at the final time $T$.
Figure 3.  Time-evolution of the exact solution at point $(x_{N_x-1},y_{N_y-1})$ in the two dimensional case (A) with $T = 12$ and $N_t = 501$; resp. at point $y_{N_y-1}$ in the one dimensional case with $T = 10$, $a = 0$ and $N_t = 501$ (B).
Figure 4.  Representation of the numerical solution $f^{\epsilon}$ for two values of $\epsilon$, and at the final time $T$, corresponding to the IMEX scheme.
Figure 5.  Left (A): Plot of the num. sol. $f^{\epsilon}$ for $\epsilon = 10^{-10}$, at the final time $T$. Right (B): Time-evolution of the IMEX scheme sol. at point $y_{N_y-1}$ in the 1D case for $T = 10$ and several $\epsilon$. We have added the exact solution for $\epsilon = 1$.
Figure 6.  Time-evolution of the solution via Fourier (A) and IMEX, MM- resp. Lagrange-multiplier schemes (B), at $y_{N_y-1}$ in 1D with $T = 10$, $a = 0$, $N_t = 501$. We have added in both cases the exact solution for $\epsilon = 1$.
Figure 7.  Evolution of the $L^{\infty}$-error between $f^{\epsilon}_{ex}(t,\cdot)$ and $f^{\epsilon}(t,\cdot)$ at final time $T = 1$ and for $\epsilon = 1$, as a function of $\Delta x$ (with $N_y = 15 001$, $N_t = 15 001$), $\Delta y$ (with $N_x = 15 001$, $N_t = 15 001$) and $\Delta t$ (with $N_x = N_y = 1 001$).
Figure 8.  Evolution of $\eta_\epsilon(T)$ and $\gamma_\epsilon(T)$ as a function of $\epsilon$ for each scheme.
Figure 9.  Condition number $cond(A)$ as a function of $\epsilon$ in log-log scale. The three curves correspond to the IMEX, Micro-Macro and Lagrange-multiplier schemes.
Figure 10.  Condition number $cond(A)$ as a function of $\epsilon$ in log-log scale. The two curves correspond to the IMP and Lagrange-multiplier schemes.
Figure 12.  Representation of the function $f^{\epsilon}$ at the final time $T$ for the IMP and Lagrange-multiplier scheme, with several values of $\epsilon$.
Figure 11.  Representation of a cut at $x = 0$ of $f^{\epsilon}_{num}$ at the final time $T$ for the IMP and Lagrange-multiplier schemes, and several values of $\epsilon$.
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